| L(s) = 1 | + (−1.85 − 0.741i)2-s + (−2.48 − 1.67i)3-s + (2.90 + 2.75i)4-s + (1.04 − 5.95i)5-s + (3.37 + 4.96i)6-s + (−1.18 − 1.41i)7-s + (−3.34 − 7.26i)8-s + (3.36 + 8.34i)9-s + (−6.36 + 10.2i)10-s + (−14.9 + 2.63i)11-s + (−2.58 − 11.7i)12-s + (−6.81 + 2.48i)13-s + (1.15 + 3.50i)14-s + (−12.5 + 13.0i)15-s + (0.820 + 15.9i)16-s + (2.22 − 3.85i)17-s + ⋯ |
| L(s) = 1 | + (−0.928 − 0.370i)2-s + (−0.828 − 0.559i)3-s + (0.725 + 0.688i)4-s + (0.209 − 1.19i)5-s + (0.562 + 0.826i)6-s + (−0.169 − 0.201i)7-s + (−0.417 − 0.908i)8-s + (0.374 + 0.927i)9-s + (−0.636 + 1.02i)10-s + (−1.36 + 0.239i)11-s + (−0.215 − 0.976i)12-s + (−0.524 + 0.190i)13-s + (0.0824 + 0.250i)14-s + (−0.839 + 0.869i)15-s + (0.0512 + 0.998i)16-s + (0.130 − 0.226i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 - 0.378i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0497958 + 0.253551i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0497958 + 0.253551i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.85 + 0.741i)T \) |
| 3 | \( 1 + (2.48 + 1.67i)T \) |
| good | 5 | \( 1 + (-1.04 + 5.95i)T + (-23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (1.18 + 1.41i)T + (-8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (14.9 - 2.63i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (6.81 - 2.48i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-2.22 + 3.85i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (30.1 - 17.4i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-8.19 + 9.77i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (26.3 + 9.59i)T + (644. + 540. i)T^{2} \) |
| 31 | \( 1 + (-34.0 + 40.5i)T + (-166. - 946. i)T^{2} \) |
| 37 | \( 1 + (-1.20 + 2.08i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (44.2 - 16.1i)T + (1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-15.3 + 2.71i)T + (1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (-13.3 - 15.8i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 + 77.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-69.2 - 12.2i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-25.1 + 21.0i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (25.3 + 69.6i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (81.7 + 47.2i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (42.1 + 72.9i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-0.961 + 2.64i)T + (-4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-19.3 + 53.0i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-55.7 - 96.5i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (13.0 + 73.9i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.77325647438005639735161289190, −11.87018341306556292224805425003, −10.65463127108044890450802548388, −9.824008948503641440690483852686, −8.415405807476753141518175985165, −7.55433316746914349159333399414, −6.13133128304892881160742076906, −4.69113620833712236856052700475, −2.05833438632981511518122591590, −0.26192830753461030908701017946,
2.76068737908701733278953715899, 5.17783003468857675170305236643, 6.33866021430089037830500501221, 7.24604253542275112685683104785, 8.756960172362105111490234165727, 10.16801357008633032921781732129, 10.55941036521967862107656904237, 11.42941392210786554546593957111, 12.87083811488190798356013932925, 14.52761517595628642045745037108
